Better-balanced-pair inequality conjecture for the cd-index

The conjecture is false for $cd$-monomials of degree $25$.

Use the list notation

$$(a_1,\ldots,a_r)=c^{a_1}dc^{a_2}d\cdots dc^{a_r},$$

and take

$$M=(0),\qquad L=(0,7),\qquad N=\varnothing,$$

together with

$$(m_1,n_1)=(5,5),\qquad(m_2,n_2)=(4,6).$$

Both pairs have sum $10$, and

$$|5-5|=0<2=|4-6|,$$

so $(5,5)$ is strictly better balanced than $(4,6)$. The conjecture therefore predicts

$$\beta(0,5,0,7,5)>\beta(0,4,0,7,6). \tag{1}$$

The coefficients can be computed directly from the established Lemmas 3.2 and 4.4 of the source. For every nonempty list $a=(a_1,\ldots,a_r)$, these give

$$\begin{aligned} \beta(a) ={}&\sum_{\substack{1\leq i\leq r\a_i>0}} \beta(a_1,\ldots,a_i-1,\ldots,a_r)\\ &+\sum_{i=1}^{r-1} \beta(a_1,\ldots,a_{i-1},a_i+a_{i+1}+1, a_{i+2},\ldots,a_r), \end{aligned} \tag{2}$$

with initial condition $\beta((0))=1$. Every list on the right has degree one less than $a$, where

$$\deg(a)=\sum_i a_i+2(r-1),$$

so (2) is a finite exact integer recursion.

Evaluating (2) gives

$$\beta(0,5,0,7,5)=1{,}287{,}894{,}857{,}340,$$

whereas

$$\beta(0,4,0,7,6)=1{,}288{,}645{,}806{,}880.$$

Thus

$$\beta(0,5,0,7,5)-\beta(0,4,0,7,6) =-750{,}949{,}540<0,$$

which is the opposite of (1).

As an independent coefficient recurrence, Proposition 2 gives

$$\Psi(B_{r+1})=\Psi(B_r)c+D(\Psi(B_r)), \qquad D(c)=d,\quad D(d)=cd,$$

with $D$ a derivation. Therefore $b(\varnothing)=1$ and

$$b(w) =\mathbf1_{\{w\text{ ends in }c\}}b(w\text{ with its last }c\text{ removed}) +\sum_{\text{$d$-positions}}b(w\text{ with that }d\text{ replaced by }c) +\sum_{\text{$cd$-substrings}}b(w\text{ with that }cd\text{ replaced by }d).$$

This separate recursion yields the same two values. Hence the more balanced pair gives a strictly smaller Boolean-lattice $cd$-coefficient, disproving the conjecture.